Abstract

Astrophysical radiation sources are scrutinized in search of
superluminal γ-rays. The tachyonic spectral densities generated by
ultra-relativistic electrons in uniform motion are fitted to the
high-energy spectra of Galactic supernova remnants, such as RX
J0852.0−4622 and the pulsar wind nebulae in G0.9+0.1 and MSH 15-52. The
superluminal spectral maps of the unidentified TeV γ-ray sources HESS
J1303−631, TeV J2032+4130 and HESS J1825−137 are inferred from EGRET,
HEGRA and HESS data. Tachyonic cascade spectra are quite capable of
generating the spectral curvature seen in double-logarithmic plots, as
well as the extended spectral plateaus defined by EGRET flux points in
the GeV band. The curvature of the TeV spectra is intrinsic, caused by
the Boltzmann factor in the source densities. The spectral averaging
with thermal and exponentially cut power-law electron densities can be
done in closed form, and systematic high- and low-temperature
expansions of the superluminal spectral densities are derived.
Estimates on the electron/proton populations generating the tachyon
flux are obtained from the spectral fits, such as power-law indices,
temperature and source counts. The cutoff temperatures of the source
densities suggest ultra-high-energy protons in MSH 15-52, HESS
J1825−137 and TeV J2032+4130.

1. Introduction

The traditional way to introduce faster-than-light particles
(tachyons) is to start with the Lagrangian

(1.1)

ηαβ = diag(−1, 1, 1, 1),
mt > 0,
which differs from the Lagrangian of a classical subluminal particle
just by a minus sign under the root [1], [2], [3] and [4]. The
superluminal particle is coupled by minimal substitution to the
electromagnetic field as indicated, if it carries electric charge e.
In this article, a very different approach to superluminal signals is
investigated, a Proca equation with negative mass-square, very contrary
to the prevailing view of tachyons as electrically charged point
particles (1.1). The Proca
field is coupled to a current of subluminal massive particles, the
Lagrangians of the superluminal radiation field and a subluminal
classical particle coupled to this field read

(1.2)

(1.3)

respectively, where mt is
the mass of the Proca field Aα.
The mass term in (1.3) is added
with a positive sign, so that
is the negative mass-square of the tachyonic radiation field. The
subluminal particle of mass m as defined by (1.3)
is supposed to carry tachyonic charge q, analogous
to but independent of electric charge, by which it couples to the
tachyon potential via a current j0 = ρ = qδ(x−x(t)),
j = qvδ(x−x(t)).
Evidently, the Proca Lagrangian is designed in analogy to
electrodynamics, but is otherwise unrelated to electromagnetic fields,
and the real vector potential Aα
is itself a measurable quantity, as the mass term breaks the gauge
invariance. Tachyons emerge as an extension of the photon concept, a
sort of photons with negative mass-square [5] and [6]. The
superluminal radiation field does not carry any kind of charge,
tachyonic charge q is a property of subluminal
particles, as is electric charge. In the geometrical optics limit of
this field theory, one can still describe superluminal rays by the
Lagrangian (1.1) with the
interaction term dropped, but otherwise the new concept is subluminal
source particles emitting superluminal radiation. The negative
mass-square thus refers to the radiation rather than the source, in
strong contrast to the traditional approach based on superluminal
source particles emitting electromagnetic radiation [1] and [4].

At first sight, the formalism closely resembles
electrodynamics, but there are differences as well. Apart from the
superluminal speed of the quanta, the radiation is partially
longitudinally polarized [7]. A basic
feature of photons traveling over cosmological distances is their
ability to propagate dispersion free, as the wave fields are
conformally coupled to the background metric. This property is also
retained for tachyons, as the tachyon mass scales inversely with the
cosmic expansion factor [5]. There is no
interaction of tachyons with photons, they couple only indirectly via
matter fields. Therefore, contrary to electromagnetic γ-rays,
high-energy tachyons cannot interact with the infrared background
radiation, so that there is no attenuation of the extragalactic tachyon
flux due to electron–positron pair creation.

The most pronounced difference between electromagnetic and
tachyonic radiation lies in the emission process itself. Freely moving
electrons can radiate superluminal quanta. Radiation densities attached
to a uniformly moving charge have no analog in electrodynamics, their
very existence has substantial implications for the space–time
structure, requiring an absorber that breaks the time symmetry of the
emission process [8]. The
tachyonic radiation densities depend on the velocity of the uniformly
moving charge in the rest frame of the cosmic absorber, and there is a
radiation threshold, a minimal speed for superluminal radiation to
occur [9]. The Green
function of this radiation process is time-symmetric, there is no
retarded propagator supported outside the lightcone. The advanced modes
of the radiation field are converted into retarded ones by virtue of a
nonlocal, instantaneous interaction with the cosmic absorber medium [10]. The latter
also defines a universal frame of reference necessary to render the
superluminal signal transfer causal. The time symmetry of the Green
function implies that there is no radiation damping, the radiating
charge stays in uniform motion. The energy radiated is supplied by the
oscillators constituting the absorber medium.

The quantization of superluminal field theories has constantly
been marred by the fact that there is no relativistically invariant way
to distinct positive and negative frequency solutions outside the
lightcone, a consequence of time inversions by Lorentz boosts. When
attempting quantization, the result was either unitarity violation or a
non-invariant vacuum state [2] and [4]. Therefore,
relativistic interactions of superluminal quanta with matter have never
been worked out to an extend that they could be subjected to test. In
fact, tachyons have not been detected so far, and one may ask why.
There is the possibility that superluminal signals just do not exist,
the vacuum speed of light being the definitive upper bound. In an open
universe, however, this is not a particularly appealing perspective.
There is another explanation, the interaction of superluminal radiation
with matter is very weak, the quotient of tachyonic and electric fine
structure constants being αq/αe ≈ 1.4 × 10−11,
and therefore superluminal quanta are just hard to detect [5]. There have
been searches for superluminal particles, which were assumed to be
electrically charged sources of vacuum Cherenkov radiation, and
bubble-chamber events were reanalyzed in search of negative
mass-squares inferred from energy–momentum conservation [11]. Apart from
that, tachyonic quanta should have been detected over the years,
accidentally, despite of their tiny interaction with matter. The most
likely reason as to why this has not happened is this: Due to our
contemporary spacetime conception, we are obliged to systematically
ignore them.

In fact, in a relativistic spacetime view, superluminal
signals are causality violating, irrespectively of the special
mechanism of the transmission. If two events are connected by a
tachyonic signal, a Lorentz boost can interchange the time order, so
that some observers will see the effect preceding the cause, absorption
preceding emission, for instance [4]. Hence,
outside the lightcone, in the domain of tachyonic signal transfer,
there is no relativistically invariant meaning to cause and effect.
Clearly, there are acausal solutions in electrodynamics as well,
advanced wave fields, and even in Newtonian mechanics one can easily
specify acausal initial conditions such as a negative time delay.
However, these solutions can be identified and discarded on the grounds
of causality violation. When superluminal signals are involved in a
relativistic context, this cannot be done, as there is no invariant way
to distinct causal from acausal solutions. What appears causal to one
observer is acausal to others, since Lorentz boosts can invert the time
order of spacelike connections. This is in strong contrast to the
retarded wave fields of electrodynamics, which stay retarded in all
frames. Retarded superluminal waves, however, will usually acquire an
advanced acausal component if subjected to Lorentz boosts. Superluminal
signals are thus incompatible with the relativity principle, as they
unavoidably entail causality violation, quite independently of the
physical mechanism of the signal transfer [5]. This
suggests to use an absolute spacetime conception when dealing with
tachyons. Relativity theory is a theory of subluminal motion, and the
extension of the relativity principle to spacelike connections
conflicts with causality. To reconcile superluminal signals with the
causality principle, it is necessary to find an absolute time order as
defined by cosmic time in the comoving galaxy frame, the rest frame of
the absorber medium. The cosmic time order unambiguously determines the
causality of events connected by superluminal signals. All observers
can arrive at the same conclusion by synchronizing their proper time
with cosmic time, irrespectively of time inversions that may occur in
individual rest frames [10]. Had there
been detection in the above mentioned searches, in the relativistic
framework in which they were interpreted, this would have been
tantamount to causality violation. By contrast, the radiation mechanism
suggested here, based on a Proca field with negative mass-square
minimally coupled to subluminal matter, is non-relativistic, as it
invokes the absolute spacetime defined by the absorber, even though the
Lagrangians are covariant.

Unitarity violation [12], an
unstable vacuum [4] and
causality violation [13] are
inextricably linked with relativistic superluminal field theories.
Unitarity violation means that the quantum mechanical probability
amplitude is not conserved, an incurable inconsistency. The term
‘unstable vacuum’ is a misnomer suggesting that there are field
theories with a stable vacuum and others with an unstable one, just
like stable and unstable dynamical systems. An unstable vacuum,
however, has nothing to do with dynamical instability. It is an
inconsistency arising in quantization attempts of relativistic
superluminal theories and implies an energy functional unbounded from
below, so that one ends up with a finite system from which an infinite
amount of energy can be extracted, a perpetuum mobile. Indefinite
energy functionals can be traced back to causality violation endemic in
relativistic theories of superluminal signal transfer. Attempts to cure
the problem of negative energies in terms of an antiparticle
reinterpretation analogous to the Dirac Hamiltonian have failed, as
causality remains violated [4]; the
indefinite classical energy functional of the Dirac theory is not
related to causality violation, after all.

The technical reason why it is not possible to consistently
define a positive definite energy functional in a relativistic
superluminal field theory is elementary and does not depend on the
specific modeling of the signal transfer. Lorentz boosts can change the
sign of the energy component of a spacelike 4-momentum, so that
negative frequency modes are generated, which can be made arbitrarily
large by choosing a suitable boost. This is also the root of causality
violation; Lorentz boosts can change the time order of spacelike
connections, so that one can always find inertial frames in which the
effect precedes the cause.

It has been argued that causality-violating signal transfer
outside the lightcone is not a logical inconsistency [13], implying
that the only way to figure out whether tachyons comply with the
causality principle is to find them and to settle this empirically.
This is presumably true, but the trouble here is that relativistic
acausal theories have been very inefficient with regard to suggestions
where to look for tachyons. Causality may not be a logical necessity,
but it is a practical one regarding physical modeling. Without
supposition of the causality principle (that is, every effect has a
cause, the cause precedes the effect, and the terms cause and effect
can be invariantly attached to the respective events) one will most
likely spend one’s time solving causality paradoxa [14] rather than
calculating cross sections, the latter being indispensable for
quantitative suggestions as to where to find tachyons.

A consistent quantization, implying causality, a stable vacuum
and a unitary scattering matrix, can be achieved if we refrain from
modeling superluminal signals on the basis of a local relativistic
spacetime and invoke an absolute time order instead when dealing with
spacelike momenta. The criterion for the usefulness of any given
spacetime conception, the use of an expanding 3-space in cosmology to
model redshifts for instance, is efficiency regarding physical modeling
in a given context. A spacetime concept has no physical reality by
itself and has to be adapted to the context, not the other way round.
We employ an absolute spacetime for the modeling of superluminal signal
transfer, since an extension of the relativistic spacetime concept to
spacelike momenta is counterproductive in dealing with superluminal
signals.

A relativistic spacetime conception has at its core the
Galilean assertion of the equivalence of uniform motion and rest. The
laws of physics are the same in all inertial frames, the latter are
considered equivalent and synchronized by a group of coordinate
transformations such as Lorentz boosts. In particular, there is no
distinguished state of rest, the substitute for this is uniform
inertial motion. One can even single out a preferred frame, such as the
comoving galaxy frame defining cosmic time, this is still a
relativistic concept.

An absolute space is defined by a microscopic space structure
replacing the purely geometric Euclidean space concept of coordinate
axes labeling the void. We assume that this space structure is a
Wheeler–Feynman absorber medium [8] capable of
turning advanced superluminal wave modes into retarded ones. The
absorber defines the frame of absolute rest coinciding with the
comoving galaxy frame, so that cosmic time provides the universal time
order required in the causality principle. The absolute spacetime
concept is centered at the state of rest, accelerated and inertial
frames are based on this reference frame. The superluminal spectral
densities studied in Section 2 are determined by the velocity of the
radiating charge. This is not a relative velocity, it stands for the
absolute motion of the charge in the absorber medium.

The absorber medium reminds us of the prerelativistic ether
outlawed by relativity theory [15];
Michelson–Morley cannot be invoked if the absorber medium does not
affect electromagnetic wave propagation. When contemplating a
microscopic space structure, one has to keep in mind that it has to
compete with Euclidean space, already a cubic lattice structure makes
physical modeling much harder. The question thus arises how to define
this space structure without compromising the efficiency of Euclidean
modeling. The way to proceed is suggested by the Mach principle and the
Wheeler–Feynman absorber theory.

The Mach principle tries to explain the inertial force in
Newton’s equations by a nonlocal interaction with the mass content of
the universe. In practice, it is not really necessary to know the
details of this interaction, as the inertial force defines itself
locally by mass and acceleration. The Wheeler–Feynman absorber theory
was designed in the context of a time symmetric formulation of
electrodynamics [8]. In this
theory, the absorber comprises the collection of electric charges in
the universe, its interaction with time symmetric electromagnetic
fields is instantaneous and nonlocal and turns advanced wave modes into
retarded ones. Here again, it is not necessary to quantify the details
of this interaction such as the actual charge distribution, as the time
symmetric Green function and the retarded and advanced wave fields
generated are determined by the local Hamiltonian. A similar absorber
concept is used for tachyonic wave propagation. Outside the lightcone,
there are no retarded or advanced Green functions, the time symmetry of
the wave propagation is broken by the absorber medium. One can give an
explicit model of the absorber medium, in terms of uniformly
distributed microscopic oscillators interacting with the time symmetric
wave field [16], but this
is not really necessary, as the time symmetric Green function can be
calculated from the local Hamiltonian [10].

It is the interaction mechanism with matter rather than the
space and causality conception that provides the clues where to search
for tachyons. We work out specific examples, scrutinizing astrophysical
radiation sources for superluminal γ-rays. We perform tachyonic
spectral fits to the spectra of Galactic supernova remnants and other
recently discovered TeV γ-ray sources. In Section 2, we explain
the transversal and longitudinal spectral densities generated by
uniformly moving charges as well as the spectral averaging with thermal
and power-law electron distributions. These spectral densities have no
analog in electrodynamics; tachyonic synchrotron densities reduce to
these densities in the limit of infinite bending radius [17], but
electromagnetic synchrotron radiation vanishes in this limit. In
Section 3, we derive the
high- and low-temperature expansions of the averaged radiation
densities. Tachyonic spectral maps of the supernova remnants G0.9+0.1,
RX J0852.0−4622 and MSH 15-52 as well as the unidentified TeV γ-ray
sources HESS J1303−631, TeV J2032+4130 and HESS J1825−137 are obtained
by fitting EGRET, HEGRA and HESS flux points. We discuss spectral
peaks, breaks, slopes, and the curvature of tachyonic cascade spectra.
In Section 4, we compare
the cutoff temperature of the electronic/protonic source populations to
the break energies in the cosmic-ray spectrum, and point out the
possibility of ultra-high-energy protons in at least three of the
studied TeV sources. In Appendix A, we calculate
the normalization factors of the spectral densities; the singular
high-temperature expansions at integer electronic power-law index are
dealt with in Appendix B.

Only frequencies in the range 0 < ω < ωmax(γ)
can be radiated by a uniformly moving charge, the tachyonic spectral
densities pT,L(ω)
are cut off at the break frequency ωmax.
The radiation condition on the electronic Lorentz factor is γ > μ,
cf. (2.3); since ωmax(μ) = 0,
there is no emission by uniformly moving charges with Lorentz factors γμ.
The threshold on the speed of the charge for radiation to occur is thus
υ > υminmt/(2mμ).
The units = c = 1
can easily be restored. We use the Heaviside–Lorentz system, so that αq = q2/(4πc) ≈ 1.0 × 10−13
is the tachyonic fine structure constant. The tachyon mass is mt ≈ 2.15 keV/c2.
These estimates are obtained from hydrogenic Lamb shifts [5]. The
tachyon–electron mass ratio, mt/m ≈ 1/238,
gives υmin/c ≈ 2.1 × 10−3.

The spectral densities (2.1) are
generated by a Dirac current. The spectral densities of a classical
point charge are recovered by putting all mt/m-ratios
in (2.1), (2.2) and (2.3) equal to
zero, in particular,
and ,
and the classical spectral cutoff occurs at .
In the ultra-relativistic limit, γ 1, we can likewise drop all terms
containing mt/m-ratios,
so that the ultra-relativistic quantum densities coalesce with the
classical densities. Substantial quantum effects regarding the shape of
the spectral density emerge only in the extreme non-relativistic limit [9]. We may then
parametrize γ = μ + ε,
with ε 1,
so that the source velocity in the vicinity of υmin
reads

(2.4)

and we may substitute
in (2.1) and (2.3). If the
polarization is not observed, we use the total spectral density pT+L(ω) pT(ω) + pL(ω).

The radiation densities (2.1) refer to
single charges with Lorentz factor γ. We average
them with electron distributions, power-laws exponentially cut with
Boltzmann distributions, dρ ∝ E−2 − αe−E/(kT)d3p,
or

(2.5)

where βmc2/(kT).
The electronic Lorentz factors range in an interval γ1γ < ∞,
the lower edge satisfies the radiation condition γ1μ,
cf. after (2.3). Linear
combinations of (2.5) may be
used for wideband spectra, including pure power-laws, cf. the modeling
of γ-ray burst spectra [18] and [19] and
electron distributions in galaxy clusters [20]. The
normalization factors Aα,β(γ1,n1)
of densities (2.5) are
determined by ,
where n1 is the electron
count and γ1 the smallest
Lorentz factor in the source population.

The averaging is carried out via

(2.6)

with ωmax(γ)
in (2.3). These
averages can be reduced to the spectral functions

The spectral range of the radiation densities (2.1) is
0 < ω < ωmax(γ),
cf. (2.3).
Inversely, the condition
or

(2.10)

defines the minimal electronic Lorentz factor for radiation at this
frequency. (
stands for the rescaled frequency ω/mt.)
That is, an electron in uniform motion can radiate at ω
only if its Lorentz factor exceeds .
The lower edge of Lorentz factors in the electron distribution defines
the break frequency, ω1ωmax(γ1),
or

(2.11)

which separates the spectrum into a low- and high-frequency band. We
have ,
and
if ω > ω1;
γ1 = μ
corresponds to ω1 = 0.
The averaged energy density (2.6) can be
assembled as

(2.12)

with
in (2.10). The
superscripts T and L denote the transversal and longitudinal
polarization components, pTα,β
is the energy transversally radiated per unit time and unit frequency.

3. Tachyonic spectral averages and spectral
fits

To get an overview of the averaged spectral densities (2.12), we
consider their asymptotic limits. The low-temperature expansion, β 1, is readily found by substituting the
asymptotic expansion of the incomplete Γ-function [21] into the
spectral function BT,L(ω;γ;α,β),
cf. (2.8),

(3.1)

where

(3.2)

This holds for Lorentz factors γμ,
cf. after (2.5),
is defined in (2.9), and the
expansion parameter is βγ 1.

The high-temperature expansion, β 1,
of BT,L is obtained by
substituting the ascending series [21]

(3.3)

The expansion applicable for βγ 1
is thus

(3.4)

with the shortcut

(3.5)

In the opposite limit, βγ 1, we may still use the low-temperature
expansion (3.1), even if β 1.
Singularities occurring at integer α in (3.4) and (3.5) cancel if ε-expanded.
To this end, we substitute α = n − ε
as well as the ε-expansion of the Γ-function
at its poles,

If γ1 = μ,
β 1
and α > 1, the maximum of pT,L(ω; γ1, n1)α,β,
cf. (2.12), is
determined by the leading factor
in the high-temperature expansion (3.4). This peak
is located at
and is followed by power-law decay ∝ ω−α
for ωmt/β
and exponential decay starting at about ω ≈ mt/β.
The low-temperature expansion (3.1) applies
for ωmt
/β. If γ1 1 (but still βγ1 1
and α > 1), the peak of pT,L(ω; γ1, n1)α,β
is determined by the factor
in (3.4) and occurs
at ωmax ≈ mt.
It is followed by a broken power-law, at first pT,Lα,β ∝ 1/ω
in the range 1 ω/mtγ1
and then pT,Lα,β ∝ ω−α
in the interval γ1ω/mt 1/β.
At ω ≈ mt/β,
there is an exponential cutoff according to (3.1).

where d is the distance to the source. The
preceding scaling relations for pT,Lα,β
give a plateau value E2dNT,L/dE ∝ 1
in the range 1 E/(mtc2) γ1,
followed by power-law decay E2dNT,L/dE ∝ E1 − α
in the band γ1E/(mtc2) 1/β,
and subsequent exponential decay.

Fig. 1. Spectral map of the pulsar wind nebula in SNR
G0.9+0.1. Data points from HESS [22]. The solid
line T+L depicts the unpolarized differential tachyon flux dNT+L/dE,
rescaled with E2 for better
visibility of the spectral curvature, cf. (3.10). The
transversal (T, dashed) and longitudinal (L, dotted) flux densities dNT,L/dE
add up to the total flux T + L. The
high-temperature/low-frequency approximation (dot-dashed) of
T + L is based on the first two orders of the
ascending series in (3.4), the
low-temperature/high-frequency asymptotics (dot-dot-dashed) is
calculated from the two leading orders in (3.1). The
unpolarized flux T + L is the actual spectral fit,
the parameters of the electron density are recorded in Table 1.

Fig. 2. Spectral map of SNR RX J0852.0−4622, data
points from CANGAROO-II [25] and HESS [26]. The plots
are labeled as in Fig. 1. T and L
stand for the transversal and longitudinal flux components, and T+L
labels the total unpolarized flux. Only the first two CANGAROO-II
points (above the 10−11-mark) have been taken
into account in the spectral fit. The CANGAROO and HESS data sets
define different slopes, the CANGAROO slope is much steeper. γ-Ray
observations of the Galactic center [27] and [28] also
resulted in diverging slopes, cf. Fig. 4 in [28]. The HESS
points and the two overlapping CANGAROO points define an extended
spectral plateau in the high GeV range typical for cascade spectra,
which is followed by exponential decay. The parameters of the inferred
thermal electron density are listed in Table 1.

Fig. 3. Spectral map of the extended TeV γ-ray source
HESS J1303−631. Data points from HESS [35]. The
unpolarized spectral fit T + L is based on the
electron density in Table 1, the
polarized flux components are labeled T and L. The high- and
low-temperature approximations of T + L are indicated
by truncated dot-dashed and double-dot-dashed curves, respectively,
with cross-over in the low TeV range. The electron populations
producing the spectra in Fig. 1, Fig. 2 and Fig. 3 are
thermal, the electron temperature is of the same order in all three
sources, cf. Table 1. The
spectral curvature is generated by the Boltzmann factor, cf. (3.1), the GeV
plateau terminates in exponential decay without a power-law cross-over.

Fig. 4. Spectral map of the pulsar wind nebula in MSH
15-52. Data points from HESS [37]. The
tachyonic spectral fit (solid line T+L) is obtained with a non-thermal
electron distribution, cf. Table 1. The fit
is for unpolarized radiation and can be split into a transversally (T)
and longitudinally (L) polarized flux. The cutoff frequency of the
averaged radiation densities (T, L and T+L) is Ecut = (mt/m)kT ≈ 70 TeV,
defined by
according to the low-temperature/high-frequency expansion (3.1). The high-
and low-temperature asymptotics of the unpolarized total flux T+L is
depicted by the dot-dashed and dot-dot-dashed curves. Most of the data
points lie in the high-temperature/low-frequency regime, contrary to Fig. 1, Fig. 2 and Fig. 3. The
decay in this regime is gradual but not power-law, with electron index α < 1,
cf. the discussion following (3.10).

In the foregoing discussion of the high-temperature
asymptotics, we assumed an electron index α > 1.
We still have to settle the case α < 1.
The spectral peak at ωmax ≈ mt
is again determined by the factor .
Adjacent is a power-law slope pT,Lα,β ∝ 1/ω
in the range 1 ω/mt 1/β,
so that E2dNT,L/dE ∝ 1,
cf. (3.10). At ω ≈ mt/β,
there is the cross-over to exponential decay, cf. (3.1). This
holds true for γ1 = μ,
cf. (2.3), as well
as γ1 1, provided βγ1 1,
since the leading order of (3.4) does not
depend on γ for α < 1.
If βγ1 1 despite β being
small, then the low-temperature expansion (3.1) applies
instead of (3.4).

As the polarization is not observed, we add the transversal
and longitudinal densities, writing pT+LpT + pL for
the energy density and NT+LNT + NL
for the number flux, the spectral fits in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5 and Fig. 6 are based
on the unpolarized differential flux, dNT+L/dE,
defined in (3.10). We
restore the natural units on the right-hand side of (3.10), up to
now we have used = c = 1,
and write E(1) for ω(1);
the plots in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5 and Fig. 6 are in
TeV units, with dNT,L/dE
in units of TeV−1s−1cm−2.
The normalization factor in (2.12) is
dimensionless, ,
where n1 is the number of
radiating electrons with Lorentz factors exceeding γ1,
distributed according to density dρα,β
in (2.5). α
is the electronic power-law index, and β the
exponential cutoff in the electron spectrum, both to be determined from
the spectral fit like n1 and
γ1. As for the electron count
n1, it is convenient to use a
rescaled parameter
for the fit,

(3.11)

which determines the amplitude of the tachyon flux. Here, [keV
s] implies the tachyon mass in keV units, that is, we put mtc2 ≈ 2.15
in the spectral function (2.8). The
tachyon–electron mass ratio is mt/m ≈ 1/238,
cf. after (2.3). The
fitting procedure and in particular the spectral maps in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5 and Fig. 6 are not
affected by the often uncertain distance estimate, in contrast to the
source count, cf. Table 1.

The electron densities dρα,β
in (2.5) are defined by parameters (α, β,
γ1, )
extracted from the spectral fit. α is the
electronic power-law index and β the cutoff
parameter in the Boltzmann factor. The lower edge of electronic Lorentz
factors in the source population is identified with the radiation
threshold, γ1 = μ,
cf. (2.3), except
for HESS J1825−137 and TeV J2032+4130.
is the amplitude of the tachyonic flux density, from which the electron
count ne is inferred, cf.
(3.11). d is the distance to the source, and kT
the electron temperature. The high-energy spectra of the SNRs G0.9+0.1
and RX J0852.0−4622 as well as the extended TeV source HESS J1303−631
are thermal, α = −2. The spectra
of the pulsar wind nebula MSH 15-52 and the TeV sources HESS J1825−137
and TeV J2032+4130 are power-law with exponential cutoff. The distance
estimates do not affect the spectral maps in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5 and Fig. 6 but the
electronic source count ne.
The distances of RX J0852.0−4622, HESS J1303−631 and TeV J2032+4130
could be one order lower than the listed ones, references are given
after (3.11).

Fig. 1 shows the
high-energy spectral map of the composite SNR G0.9+0.1 [22], [23] and [24] in the
Galactic center, at a distance of 10 kpc [23] and [24]. The core
component is presumably a pulsar wind nebula, though no pulsed emission
has yet been detected.

In Fig. 2, we show
the tachyonic spectral map of the shell-type SNR RX J0852.0−4622 [25], [26], [27], [28], [29], [30], [31], [32], [33] and [34]. The
1.157 MeV line of the short-lived 44Ti
isotope (with a half-life of 60 yr) was detected by COMPTEL,
and an age of 700 years and a distance of 0.2 kpc was inferred
[29], [30] and [34]. This
detection, however, was marginal [32], and column
density estimates suggest a distance of 1–2 kpc, in the
direction of and probably associated with the Vela Molecular Ridge [30]. A compact
X-ray source has been located near the SNR center [32] and [33].

The spectrum of the extended TeV source HESS J1303−631 [35] is depicted
in Fig. 3. The
distance of 15.8 kpc is inferred from an association with PSR
J1301−6305 [36], but could
be one order smaller, since associations with other pulsars or
molecular clouds at much smaller distance are possible [35].

In Fig. 4, we plot
the spectral map of the pulsar wind nebula in MSH 15-52 [37], [38], [39] and [40]. The nebula
is powered by PSR B1509−58 [38] and [39], located at
a distance of 4.4 kpc [40].

The spectrum of the TeV source HESS J1825−137 [41], [42], [43] and [44] is shown in
Fig. 5. This
source is associated with the EGRET source 3EG J1826−1302 [41], which is
believed to be powered by PSR B1823−13 [43]. A further
association of this pulsar with HESS J1825−137 is suggested by the
one-sided X-ray morphology of the asymmetric pulsar wind nebula
G18.0−0.7 powered by PSR B1823−13 [44], since the
TeV emission of HESS J1825−137 is likewise strongly one-sided [41]. The
distance estimate for PSR B1823−13 is 4.1 kpc [40].

Fig. 6 shows the
spectral map of the presumably extended TeV source TeV J2032+4130 [45], [46], [47] and [48], associated
with the EGRET source 3EG J2033+4118 [45]. The
distance estimate is 15 kpc, inferred from an association with
the low-mass X-ray binary 4U 1624–49 [47] and [48]. However,
TeV J2032+4130 could be as close as 1.7 kpc if located in the
Cyg OB2 association [45] and [46].

Once
is extracted from the spectral fit, we calculate via (3.11) the
number of radiating electrons n1
constituting the density dρα,β.
This, however, implies that all tachyons reaching the detector are
properly counted. At γ-ray energies, only a tiny αq/αe-fraction
(the ratio of tachyonic and electric fine structure constants, αq/αe ≈ 1.4 × 10−11)
of the tachyon flux is actually absorbed [49] and [50]. This
requires a rescaling of the electron count n1,
so that the actual number of radiating electrons is nen1αe/αq ≈ 7.3 × 1010n1. (This rescaling applies
to γ-ray spectra only, to frequencies much higher
than the 2 keV tachyon mass, so that the mass-square can be
dropped in the tachyonic dispersion relation. At X-ray energies, we
have to rescale with the respective cross-section ratio, σe(ω)/σq
(ω), where σe
is the electromagnetic cross-section and σq
its tachyonic counterpart [7].) In Table 1, we list
the flux amplitude
inferred from the spectral fit, as well as the renormalized electron
count ne depending on the
distance estimate. The density dρα,β
is defined by parameters
obtained from the spectral fit. For instance, the source count in the
surface field of PSR B1509−58 is ne ≈ 1.3 × 1047[49], to be
compared to 8.3 × 1047
obtained here for the pulsar wind nebula in MSH 15-52, cf. Table 1.

4. Conclusion

We have studied superluminal radiation from electrons in
uniform motion. Tachyonic synchrotron and cyclotron radiation were
investigated in [17]. In the
zero-magnetic-field limit, the tachyonic synchrotron densities converge
to the spectral densities (2.1) for
uniform motion. Orbital curvature induces modulations in the spectral
densities. At μG-field strengths as encountered in
the shock-heated plasmas of SNRs, these oscillations are quite small,
just tiny ripples along the slope of densities (2.1) with
increasing amplitude toward the spectral break. If integrated over a
thermal or power-law electron population, the oscillations are averaged
out, so that the spectral densities (2.12) can still
be used for electron trajectories bent by weak magnetic fields. As
discussed in the Introduction, the source particles radiate in uniform
motion, there is no electromagnetic radiation damping unless the
trajectories spiral through magnetic fields. The energy radiated is
drained from the absorber breaking the time symmetry of the Green
function outside the lightcone.

We averaged the superluminal flux densities with electronic
source distributions, derived the high- and low-temperature expansions
of the spectral averages, and explained the spectral plateaus, the
power-law slopes and the curvature of the spectral maps. We
demonstrated that the γ-ray spectra of the
supernova remnants and TeV sources listed in Table 1 can be
reproduced by tachyonic cascade spectra. Speed and energy of tachyonic
quanta are related by ω = mtc2(υ2/c2 − 1)−1/2.
At γ-ray energies, their speed is close to the
speed of light, the basic difference to electromagnetic radiation is
the longitudinally polarized flux component. The polarization of
tachyons can be determined from transversal and longitudinal ionization
cross-sections, which peak at different scattering angles [51].

The source numbers listed in Table 1 are
lower bounds based on γ-ray emission only. The tachyonic fine structure
constant enters as a square in the number count, once via (3.11) when
inferring the preliminary count n1
from the spectral fit, and a second time when rescaling this count to
obtain the actual source number .
The tachyonic fine structure constant is an independent estimate from
Lamb shifts in hydrogenic ions [5], it enters
only linearly in the spectral density (2.1), in
contrast to the squared electromagnetic constant in the Klein–Nishina
cross section. The cutoff in the electron energy is much higher
compared to a Compton fit, which compensates the small tachyonic fine
structure constant in the spectral density and the absorption
probability, as well as the reduced source count.

Source counts based on Compton fits have not yet been
calculated for the sources discussed here, but one can obtain a rough
estimate if one considers the Compton count for the Crab Nebula and
rescales with the observed TeV power ratio. To this end, we use the
total TeV flux of the respective source in Crab units, such as F/FCrab ≈ 0.02
for SNR G0.9+0.1, cf. [22]. The flux
of SNR RX J0852.0−4622 is roughly that of the Crab [26]. A flux
ratio of 0.17 for HESS J1303−631 is quoted in [35], and a
ratio of 0.15 for MSH 15-52 in [37]. The TeV
flux of HESS J1825−137 is a 0.68 fraction of the Crab [41], and the
flux collected from TeV J2032+4130 amounts to 5% of the Crab flux [46]. The power
ratio is obtained as P/PCrab ≈ (d2[kpc]/4)F/FCrab,
with 2 kpc to the Crab. The distance estimates of the
respective sources are listed in Table 1.

A count of nCrab ≈ 1.3 × 1051
electrons/positrons in the Crab was derived from an electromagnetic
inverse-Compton model in [52]. An
estimate of the Compton count for the above sources is found by
rescaling with the power ratio. We obtain nCrabP/PCrab ≈ 3.7 × 1051
both for HESS J1825−137 and TeV J2032+4130 (based on the distance
estimates in Table 1); the
estimates for the other sources are similar within one order of
magnitude. The counts based on the tachyonic cascade fits in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5 and Fig. 6 are two
to three orders lower than the respective Compton count, cf. Table 1. In
either case, the distance estimate enters as a square, which can lower
the source count of RX J0852.0−4622, HESS J1303−631 and TeV J2032+4130
by two orders, cf. after (3.11).

Apart from the extended plateaus in the spectral maps, it is
the high cutoff in the electron energy rather than the mildly reduced
source count that distinguishes tachyonic cascade spectra from
inverse-Compton fits and, for that matter, hadronic radiation models
based on pion decay [53] and [54]. In the
case of a thermal electron distribution, with electron index α = −2,
we may identify kT[TeV] ≈ 5.11 × 10−7/β,
cf. (2.5). We use
this definition of electron temperature also for other electron
indices; kT is the energy at which the exponential
decay of the electron density sets in, cf. the end of Appendix A. The TeV
spectra discussed here show significant curvature, and we demonstrated
that this curvature is reproduced by an exponentially cut power-law
electron distribution. The exponential shows in the
low-temperature/high-frequency expansion of the averaged spectral
density, cf. (2.12) and (3.1), curving
the spectral slopes in double-logarithmic plots. This curvature is
intrinsic, there is no attenuation owing to infrared background
photons, as tachyons do not interact with electromagnetic radiation
fields. Besides, the sources discussed here are all Galactic; tachyonic
spectral maps of TeV blazars are studied in [55]. The cutoff
kT in the electron energy listed in Table 1 is to be
compared to the spectral breaks in the cosmic-ray spectrum. These
breaks occur at about 103.5, 105.8
and 107 TeV, dubbed knee, second knee
and ankle, respectively [56]. In MSH
15-52 as well as HESS J1825−137 and TeV J2032+4130, the cutoff happens
between the first and second knee. As for protonic source particles,
the cutoff energies kT in Table 1 have to
be multiplied with 1.8 × 103,
the proton/electron mass ratio, so that we arrive beyond the ankle, at
protons in the 1021 eV range, which
suggests Galactic TeV γ-ray sources as production sites of
ultra-high-energy cosmic rays. In this case, the tachyon/electron mass
ratio in the spectral densities has to be replaced by mt/mp ≈ 2.3 × 10−6,
cf.(2.1), (2.2), (2.3), (2.4), (2.5), (2.6), (2.7), (2.8), (2.9), (2.10) and (2.11). In Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5 and Fig. 6, however,
this is not necessary, since even the tachyon/electron ratio is too
small to noticeably affect the spectral maps.

Acknowledgments

The author acknowledges the support of the Japan Society for
the Promotion of Science. The hospitality and stimulating atmosphere of
the Centre for Nonlinear Dynamics, Bharathidasan University, Trichy,
and the Institute of Mathematical Sciences, Chennai, are likewise
gratefully acknowledged.

We derive the low- and high-temperature expansions of the
normalization factors Aα,β(γ1,n1)
of densities (2.5),

(A.1)

where n1 is the number of
source particles with Lorentz factors in the range γ1γ ∞.

A.1. Low-temperature expansion of Kα,β(γ1)

To obtain the low-temperature expansion of Kα,β(γ1 = 1),
we substitute γ = x + 1
in the integrand of (A.1), and then
expand in x except for the exponential. Using
term-by-term integration (via Watson’s Lemma), we readily find

(A.2)

The low-temperature expansion of Kα,β(γ1)
for γ1 > 1
is found by substituting
into (A.1), where .
We then expand the integrand in ascending powers of x,
leaving the exponential untouched, and use term-by-term integration,

(A.3)

This expansion apparently applies if ,
and can also be used for small β (high-temperature
limit) and large γ1, as long
as this condition is met. If β is large and γ1
close to one so that
is small, we have to proceed differently,

(A.4)

with the asymptotic series Kα,β(1)
in (A.2). As for kα,β(γ1),
we write γ = 1 + δx,
where δγ1 − 1,
and expand the integrand in an ascending series in x,
except for the exponential. Term-by-term integration gives a
power-series in δ,

(A.5)

where we use the shortcut

(A.6)

with positive half-integer j. This high-temperature
expansion of Kα,β(γ1),
defined by (A.4) with
series (A.2) and (A.5)
substituted, is applicable if β 1 and δγ1 − 1 1,
so that βδ is small. If βδ is
large, the expansion of Kα,β(γ1)
in (A.3) applies.
As mentioned, series (A.3) can be
used in the high-temperature regime, β 1,
provided βγ1 1.

The coefficients in (A.20) are those
of the root in (A.7). The
notation Kα,β(0)
and kα,β(∞)
symbolizes the respective integrals in (A.7) and (A.4), with the
exponential or root formally expanded. The high-temperature expansion
of Kα,β(γ1),
defined by (A.12) and (A.16), is
efficient for βγ1 1,
provided γ1 is not too close
to one. If βγ1 1, we use the expansion of Kα,β(γ1)
in (A.3) instead.
The high-temperature expansion of Kα,β(1)
is defined by (A.19), (A.20) and (A.21). There
are singularities for integer α, which cancel in
every order if ε-expanded, cf. Appendix B.

This series is recovered from (A.15), by
substituting into (A.16) the
identity [21]

(A.24)

with b = α−k.
The series (A.23) can also
directly be obtained from (A.4); we expand
the exponential and use the variable substitution y = γ2 − 1
and the integral representation

(A.25)

The calculation of Kα,β(γ1)
via (A.22) and (A.23) is
efficient for γ1 close to
one, such as γ1 = μ,
cf. after (2.5).

We discuss the location of the peak of the densities dρα,β(γ)
in (2.5). γmax is determined
as the γ > 1 zero of γ3 + (α/β)γ2−γ−(α + 1)
/β = 0. We give only the leading
orders and state the expansion parameters. If β 1, we find γmax = 1 + 1/(2β) + O(β−2).
If β 1
and α < 0, we find γmax = −(α/β)(1 + O(β2/α3)),
from the third and second order. In leading order, the ‘displacement
law’ βγmax ≈ const
holds true; the expansion parameter is stated in the preceding O-term.
If β 1
and α > 0, ,
via the zeroth and second order; this peak is followed by a power-law
slope γ−α,
extending to γ ≈ 1/β.
Finally, if β 1
and α very close to zero, we may still have β/α3/2 1, so that the indicated expansions
break down. In this case, we find γmax = β−1/3(1 + O(β2/3,
α/β3/2)),
from the zeroth and third order, where the O-term stands for a double
series in the indicated parameters.

obtained by splitting kα,β(∞) = κ1 + κ2
in (A.21). We
substitute α = m + ε,
where m is an integer at which the series become
singular. We then expand in ε, making use of the ε-expansions
of the Γ-function in (3.6) and (B.1). The ε-poles
cancel if the series are added according to (B.2). The
following expansions of Kα,β(0)
and κ1,2 are valid up to
terms of O(ε).

We start by expanding Kα,β(0)
in (B.3). We put α = 2k + ε
and find, for k 1,

(B.6)

If k 0,
this is replaced by

(B.7)

The poles at odd integers are dealt with in like manner. We put α = 2k + 1 + ε
and find, for k 0,

(B.8)

and for k −1,

(B.9)

We turn to the ε-expansion of series κ1,2
in (B.4) and (B.5). Series κ1
is singularity free and even finite at even integer α,
so that we can use (B.4) at α = 2k.
(The series terminates when the Γ-functions in the
denominator become singular.) Series κ2
is singular at even integers. We substitute α = 2k + ε
and expand, arriving at

Series κ2 is finite and
singularity free at odd integer α, so that we can
use (B.5) at α = 2k + 1.

Thus, at integer α, we can piece together
series Kα,β(1)
in (B.2) by adding
the respective series in (B.4), (B.5), (B.6), (B.7), (B.8), (B.9), (B.10), (B.11), (B.12) and (B.13). For
instance, K−2,β(1) = K2(β)/β,
with the modified Bessel function K2,
which follows by putting k = −1
and adding Kα,β(0)
in (B.7), κ1
in (B.4) (with α = 2k),
and κ2 in (B.10). There
are no singularities in the first two orders of Kα,β(1)
for α 3
and α −2,
so that we can read them off from (A.19), (A.20) and (A.21). In
between, we find the leading orders at integer α as

(B.14)

The ε-expansion of the normalization factors Kα,β(γ1)
for γ1 > 1
is recovered from the series representation (A.22) and (A.23). Series (A.23) is
singularity free at integer α, and the ε-expansion
of Kα,β(1)
has been discussed in the foregoing. The ascending series of the
hypergeometric functions in (A.23) is only
efficient for γ1 close to
one. For γ1 1, we calculate these functions via (A.24). However,
in (A.24) there
occur singularities at α−k = b = −(2m + 1),
m −1.
In this case, we calculate the hypergeometric functions in (A.23) by means
of the identity

(B.15)

At any other integer b = α − k,
we can use (A.24), as it is
singularity free.

We finally indicate the derivation of (B.15), obtained
by ε-expansion of (A.24). We start
with the series (A.17);
singularities occur at b = −(2m + 1) + ε,
m −1.
We find,

(B.16)

This settles the ε-expansion of the left-hand side
of (A.24). On the
right-hand side, we also substitute b = −(2m + 1) + ε
and expand,